We show here a 2(dlogN) size lower bound for homogeneous depth four arithmetic formulas. That is, we give an explicit family of polynomials of degree d on N variables (with N=d3 in our case) with 01 -coefficients such that for any representation of a polynomial f in this family of the form f=ijQij where the Qij's are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that the total number of monomials in all the Qij's put together must be at least 2(dlogN) .

The abovementioned family which we refer to as the Nisan-Wigderson design-based family of polynomials, is in the complexity class VNP. For polynomial families in VP we show the following: Any homogeneous depth four arithmetic formula computing the Iterated Matrix Multiplication polynomial IMMnd --- the (11) -th entry of the product of d generic nn matrices --- has size n(logn), if d=(log2n). Moreover, any homogeneous depth four formula computing the determinant polynomial Detn --- the determinant of a generic nn matrix --- has size n(logn). Our work builds on the recent lower bound results and yields an improved quantitative bound as compared to the recent indepedent work by Kumar and Saraf (ECCC TR13-181).